 Have you ever noticed that when we buy something in a shop the shopkeeper may Scan the barcode of the product using a device with a light in it. What is that? That's a laser You are watching this video on the internet right now Which helps in the transfer of information over large distances that may involve optic fiber cables with lasers You see lasers are everywhere whether we talk about laser printers laser scanners Whether we talk about data storage using CDs DVDs blu-rays Whether we talk about industrial applications of cutting and welding of materials Whether we talk about medical technologies like lasix surgery or various kinds of skin treatments Whether we talk about military applications of guidance and weapon systems whether we talk about surveillance whether we talk about Communications research and development lasers are very important in modern technology However, the physics behind lasers is very simple. It is based on a very simple and unique Atomic transition known as stimulated emission, which was suggested by Albert Einstein in 1917 But it took almost 30 years after that for somebody to figure out everything and build an actual laser So in today's video, we are going to talk about the physics behind lasers We're going to discuss a three-level laser system And then we're going to talk about the Einstein coefficients and Einstein's approach to explaining the plant's blackbody radiation spectrum Using the concept of stimulated emission. So let's begin So here we have a various kinds of atomic transitions involving emission or absorption of radiation You see in my previous lectures, we talked about the structure of the atom We saw that the atom has discrete energy levels and the atom can make transitions from one energy level to another energy level Which is associated with either an absorption or emission of a photon of sufficient frequency So we saw that these kinds of discrete energy levels are associated with discrete transitions Involving emission or absorption of photons now this leads to mainly two kinds of transitions on one hand If we have an atom which is in the ground state, then this capable of absorbing a suitable radiation and Getting excited to a higher excited state This is known as simply absorption or stimulated absorption in which an atom in a lower state Absorbs an incident radiation of sufficient energy and gets excited to a higher state Now the second case is spontaneous emission when what happens when the atom is in the higher state or in the excited state It can automatically or spontaneously Come down to its ground state with the release of an incident photon and the photon has energy Which is equal to the difference in the energies between the excited and the ground state This is known as Spontaneous emission because it happens spontaneously it usually takes Time of around 10 to the power minus 8 to 10 to the minus 12 seconds for a spontaneous emission to take place Now it was Einstein in 1917 who suggested for the very first time that apart from these two Transitions there is a third transition which is also possible and that is known as Stimulated emission so if you have an atom in an excited state then it can be stimulated by an external photon to cause a transition or a de-excitation to a ground state Releasing another photon in that process. This is known as stimulated emission Stimulated emission is caused by an external photon of sufficient frequency It is the external photon that triggers that De-excitation process of the higher excited state of the electron to the lower excited state of the electron in this process A suitable frequency of light is emitted which is in phase with the incident Photon so this process basically takes one photon But image to photons which are in phase and which have the same frequency So it is this stimulated emission which lies at the basic working mechanism of laser So first of all, what is a laser? Let's look at the full form of the laser. You see when we talk about laser It is actually an acronym which stands for light amplification by stimulated emission of radiation now The name of the laser itself contains the concept of stimulated emission and how Amplification of light happens via the stimulated emission of radiation. Now to understand what a laser is Let's try to understand some of its properties You see how is laser different from ordinary light when we talk about ordinary light It may contain a large number of wavelengths different wavelengths which are out of phase with each other That's ordinary life for you. Now you may have Monochromatic light the meaning of a monochromatic light is that it contains the same wavelengths But they may be out of phase with each other and then you can have monochromatic Coherent light which contains the same wavelength always being in phase with each other So this third category of light is what the laser consists of so first of all a laser consists of monochromatic radiation or Monochromatic light that means it contains one wavelength and All the waves of the beam of a laser are in phase. So we call it coherent So it's a coherent monochromatic light Now because the laser is monochromatic and Coherent so what we end up getting is two more properties one is that first of all it is an intense beam You see whenever you have waves which are in phase you end up getting constructive Interference of all the waves that leads to a very intense beam and also a beam that Does not diverge or it diverges Very less or against it does not diverge What does this mean? You see when we talk about normal light like let's suppose we take a torch light It diverges along its path. So it spreads out. However, when we talk about a laser It doesn't really spread that much compared to normal radiation And that's why lasers can travel vast distances without experiencing much Divergence in its beam. So these properties constitute the Characteristics of how a laser is different from ordinary light. Now the question is how can we create? laser light having these properties Using the concept of stimulated emission for that we have to understand how atomic transitions happen between different energy levels So here we have three distinct energy levels. So let me specify which one represents what so the lower energy level is the ground state the Upper energy level is the excited state This excited state is usually short-lived. So I'm going to call it a short-lived excited state and The state in the middle is a metastable state What is the difference between an ordinary excited state and a metastable state? Is that an excited state is very short-lived while a metastable state is? Long-lived or has a long lifespan compared to the ordinary excited state now ordinary excited state usually undergoes de-excitation in a time interval of around 10 to the power minus 8 to 10 to the power minus 12 seconds while Metastable state can survive for longer durations time periods of around 30-3 Seconds or something comparable to that. So how does that make a difference? So let us try to understand what happens So whenever we have first of all Absorption at let's suppose time t is equal to 0 then because of an external radiation Let's suppose an atom goes from a lower excited state to a higher excited State so let's suppose the atom goes from the ground state to the short-lived excited state When it absorbs some sort of an incident Radiation now what happens is that because the ordinary excited state is very short-lived Usually it gets de-excited to the metastable state in a time span of around 10 to the power minus 8 seconds So this may be a spontaneous emission or this may be some kind of a non-radiative emission I'm come I'm going to come to that in a moment. Now. What's going to happen is that the atom is going to now be in this particular Metastable state for a longer duration. This is where the Lasing transition happens. This is where when if we introduce a light of sufficient or suitable frequency then this transition causes a Stimulated emission where a photon is emitted which is in phase with the incident photon and this Constitutes the laser light now at this point. You might have a few questions in mind first of all Why do we need a three-level system? Why can't we have just a two-level system, right? We were talking about absorption and spontaneous emission and stimulated emission for just two levels right ground state and excited state But why are we talking about a three-level system? You see the thing is that whenever we have an incident photon Which is incident on an atom if the photon has sufficient energy Then the atom can jump to an excited state number one But the atom can also get de-excited to a ground state if an excited state already exists so an incident photon is capable of inducing both absorption and Stimulated emission now usually if you have an assembly of atoms if you have a large number of atoms then at normal Temperatures majority of the atoms are going to be in its ground state Now if I introduce an incident radiation of sufficient frequency or sufficient energy Then a large number of those atoms will jump from the ground state to the excited state Now what happens now what happens is that the excited atoms will Immediately experience a spontaneous emission and come back to its ground state because of this reason what happens is that usually it is the ground state Which is populated number one and number two Because a spontaneous emission happens in such a small time period Therefore there isn't much time for an incident photon to come and cause stimulated emission in the process So usually the rate at which the spontaneous Emission happens is far larger compared to the rate at which stimulated emission happens now When we talk about a laser we are interested in laser amplification or light amplification So the key concept of a laser is that we need to have a large number of atoms in the excited state as Compared to the ground state right if we have an equal number of atoms in the excited state an equal number of atoms in The ground state then an incident photon will cause stimulated emission, but it will also cause Absorption so there is stimulated emission and absorption. So it kind of balances out. We do not end up getting Light amplification for light amplification to be there. We need stimulated emission to dominate over the spontaneous emission and the absorption processes and because in a two level system This doesn't usually happen because it is an incident photon causing both The absorption and the stimulated emission. We never really get a situation Usually where stimulated emission dominates over the absorption or the spontaneous emission That is why the light amplification is not possible in that kind of a two-level system However, when we introduce a three-level system, then we end up getting a situation where a Metastable state exists where the atoms can come and Exists for a longer duration, which then finally comes down to the ground state via stimulated emission processes We can create a scenario in which a large number of atoms exists in the matter stable state as compared to the ground state So such kind of a system is known as a three-level laser system Now this process in which we try to increase the number of photons in the higher state as opposed to the ground state is known as population inversion In the process of population inversion, we are essentially interested in Take having a large number of atoms in the state above the ground state so that we end up getting more stimulated emission thereby causing The lacing transitions or the lacing photons to be emitted now There are different kinds of method of population inversion one of them is called optical pumping In optical pumping we essentially introduce some sort of a suitable radiation into the material Which causes the transitions from ground state to the short-lived excited state to take place Then that is followed by an immediate non-radiative transition to the metastable state where a large number of atoms exist and then if you introduce a photon of sufficient Frequency which is different from this photon you end up getting stimulated emission to happen which constitutes the laser light now Let us discuss this process in a little bit more detail, but let me first draw a couple of diagrams So here is a three-stage laser where we will discuss each of these Steps one by one. So initially we have some sort of a material which is exposed to some incident radiation Which then makes a transition from a lower state to a higher state So to achieve this kind of a population inversion. We use optical Pumping initially. So this is the first step of optical pumping to excite a bunch of atoms in the ground state To the higher excited state So the moment the atoms in the ground state absorb some kind of an incident light and Jump from the ground state to the higher excited state it instantaneously because the higher excited state is short-lived it instantaneously Transitions to the metastable state Now this transition to the metastable state from the short-lived lift higher excited state Is usually non-radiative now. What is this non-radiative? Transition so let's say that this is point number two and I'm going to call this as Point number two non-radiative transitions Which simply means that whenever an atom Gets excited or the excited it is not usually happening all the time because of electromagnetic Radiation there are other processes via which an atom can be excited or de-excited for example In this situation where an atom went from a ground state to a higher excited state It did so because of an incident radiation of sufficient energy, but in normal situations if light is not present then because of the thermal agitation of the atoms in a given gas or a solid The kinetic energy can also cause Atom to get excited to a higher energy level all right similarly Higher excited state atom can also go to a lower excited state by losing energy in the process of internal conversion so for example non-radiative transitions for example vibrational relaxation or Creation of phonons in the crystal lattice of the solid so there are other non-radiative Mechanisms through which an atom can get De-excited to a lower energy level so this is what usually happens in a three-level system that once via optical pumping The number of atoms are pushed to a higher energy state They immediately undergo non-radiative transitions and come to a metastable state now because the metastable state is Having a long lifespan now suddenly we have a large number of atoms in the metastable state thereby Achieving the population inversion that I was talking about Now if we introduce an incident photon whose frequency is different from this photon Okay, so this photon has a frequency of let's suppose new dash and now if we introduce a photon of let's suppose frequency new then All this Atoms in the excited state will make a transition from the metastable state to the ground state and in that process You end up getting a large number of same frequency waves in Phase so you have the incident photon and then you have multiple Emitted photons in phase that constitute what is known as the lacing transition? So this is known as the lacing transition So by this method we are able to achieve a population inversion And we are able to achieve a situation in which we get stimulated emission which dominates over other processes They are by creating a large number of photons of same frequency in phase that can be Released in the form of a laser beam So the construction of the laser may look something like this where the material in the laser Experiences this kind of an optical pumping via flash tube or something similar and then the spontaneous emission from the metastable State to the ground state then starts creating stimulated emissions further and these photons Go back and forth between the partially silvered mirrors and when the number of photons is sufficient enough It penetrates through the mirror and released like a laser beam that we are familiar with now the three-stage laser is not the only Method there are other kinds of laser mechanisms. We have the three-stage laser We have the four-stage laser So the four-stage laser involves four different states where one of the transitions is associated with this stimulated emission And the other transitions may be associated with some sort of a non-radiative or other kind of a phenomena The simple process of the stimulated emission performed in a very unique manner ends up creating a highly coherent highly intense Non-diverging beam that has a large scale technological implications in today's world So therefore laser is very very important and the physics of the laser however is quite simple and interesting and arises from the way Transitions happen due to the atomic structure that we have been discussing in my previous lectures. Now, let us look at Einstein's suggestion and what he had to say about this stimulated emission In 1917 Albert Einstein suggested the stimulated emission between two energy levels and he used this idea to explain The Planck's distribution law for the black body spectrum Now if you remember we have talked about black body spectrum a couple of lectures back where we discussed this in detail that Materials usually have this kind of a radiant energy density or black body radiation That has a certain kind of a distribution which was finally given an explanation of by Max Planck via his Planck energy distribution function Now we can use Einstein's approach to come up with a similar kind of a distribution function for the energy density of Light emitted by materials. So for that, we are going to assume two different levels. So here we have a ground state let's suppose I'm going to call this as E naught and or e1 let's suppose ground state e1 Okay, n is equal to 1 and then we have the excited state here. I'm going to call this as e2 Okay, state n is equal to 2. So transitions between these two energy levels is possible via some kind of a suitable frequency new such that h nu is equal to the energy level difference and we can have three distinct mechanisms by which this can happen So let us first look at the absorption of a photon Now when we talk about absorption of a photon, then we are basically interested in Let's suppose some sort of an incident radiation. Let me draw this extend this a little bit further To complete the diagram So we can have some kind of an incident photon Which is absorbed by the atoms that let's suppose the incident photon has energy h nu and Because of this the atom makes a transition from e1 to e2 Now I'm interested in the rate of this absorption. All right So how can I calculate the rate of this particular absorption? Now first of all if we look at a material in let's suppose thermal equilibrium Then the material is emitting radiation It is also absorbing radiation But it is emitting and absorbing radiation in such a manner that its temperature is a constant of time So it's in thermal equilibrium. So let us suppose that the thermal energy Radiation density basically is given by you. Okay, so I'm going to say that I have energy density corresponding to the radiation Which is basically emitted by any kind of a material for a given frequency range Nu is given by you Nu. All right, so it will depend on that particular radiation density, right? Greater the radiation density experienced by the material Greater will be the absorption of the radiation for that particular frequency So if I am interested in figuring out what is the rate of this transition So if I'm interested in the rate of the transition from let's suppose 1 to 2 Then of course the rate of transition will depend upon the energy density corresponding to that particular frequency It will also depend upon maybe other parameters involving energy level E1 and E2 and I'm going to include those parameters in a constant I'm going to call that constant as capital B 1 2 this constant basically represents the probability of absorption. Okay, so here if I say okay B 1 2 this is the probability of absorption or Stimulated absorption Okay, now of course It will also depend upon how many number of atoms are present in the ground state E1 versus how many number of atoms are present in the excited state E2 at a given point in time So let's suppose that I have N1 number of atoms present in the ground state and N2 number of atoms present in the excited state So for an absorption of a photon the rate will also depend upon N1 So the rate at which the absorption of a photon of frequency new takes place will depend upon the energy density It will depend upon the probability of this kind of a stimulated Absorption and it will depend upon the number of atoms in the ground state fine perfect now. Let's look at the Emission of a photon now as I already mentioned to you the emission of a photon can take place via two mechanisms One is a spontaneous emission. The other is a stimulated emission. So in the spontaneous emission, you will have some sort of Radiation which spontaneously decays in a very short interval from E2 to E1 Now the spontaneous emission does not require any kind of an incident radiation the spontaneous emission happens Spontaneously automatically therefore it does not depend upon the energy density of frequencies present So therefore for spontaneous emission the rate is only proportional to the number of atoms present in the excited state and Of course the probability of spontaneous emission which I'm going to call as a To one so a to one basically depends upon It is a con it's a constant number that gives us an idea about the various parameters that Depend upon whether or not what is a probability that a transition from E2 to E1 is going to take place So you can think of a to one as the probability of spontaneous emission So this is essentially the rate at which spontaneous emission is happening now What about stimulated emission a stimulated emission does depend upon the energy density because it requires an incident radiation, right? So the stimulated emission Requires that there is an incident photon Because of which the transition from the higher level to the lower level takes place And this process releases an photon In phase with the incident photon. So therefore the stimulated emission does depend upon the energy density So it will depend upon energy density. Let's suppose it also depends upon the Properties of the energy levels, which we are going to call as some kind of a constant b To one let's suppose Where b to one is the probability of stimulated emission and of course it depends upon the number of atoms in the energy level e2 So it depends upon n 2 So therefore the rate at which an emission of a photon takes place So let's suppose I call this as capital N 2 to 1 is equal to sum of this, right? So it is equal to n 2 a 2 1 plus n 2 b 2 1 u nu, all right Now in thermal equilibrium the rate at which the photon is absorbed is equal to the rate at which the photon is emitted Because by definition of thermal equilibrium the system is at constant temperature So it is absorbing as much radiation as it is emitting that much radiation So in that kind of a situation both these two terms are going to be equal. So a thermal equilibrium We have the rate of the transitions from 1 to 2 is equal to the rate of the transitions from 2 to 1 So if I plug this into the equation, so I have n 1 b 1 2 u nu is equal to you have n 2 and within brackets you have a 2 1 Plus b 2 1 u nu, all right Now what we can do is we can divide the entire equation by n 2 and b 2 1 So let's divide the entire equation by n 2 and b 2 1 so n 1 upon n 2 and Here we will have b 1 2 upon b 2 1 U nu is equal to divided by n 2 so n 2 n 2 gets cancelled and divided by b 2 1 So here I have a 2 1 upon b 2 1 plus b 2 1 again gets cancelled you are left with u Nu, so I can rearrange the terms here. So if I bring this so if I bring this this side It simply ends up becoming u nu becomes a constant and in the bracket We have n 1 upon n 2 b 1 2 upon b 2 1 minus 1 close bracket is equal to a 2 1 upon b 2 1 So let me rewrite this equation at the top here. So then u nu Does becomes if I make a comparison u nu therefore becomes so in the numerator we have a 2 1 upon b 2 1 Okay, and then in the denominator, I have n 1 upon n 2 times b 1 2 upon b 2 1 upon b 2 1 minus 1 so this is the basically energy density as Predicted by the Einstein's approach where we have included this kind of phenomena of stimulated emission Now we can actually calculate the number of atoms at any given energy level Based on the temperature of the system because we can apply the Maxwell Boltzmann distribution function To get an idea about the number of atoms at energy level 1 and 2 because the Maxwell Boltzmann distribution function is a function That gives us an idea of the number of particles In an ensemble or in an assembly having energy e so this is basically equal to some constant Let's suppose. I call it capital a or c whatever you want to call it e to the power minus Epsilon upon k t where k is the Boltzmann constant and t is the temperature So this is a distribution function that we can use to calculate the numbers n 1 and n 2 So I can say that n 1 is equal to a e to the power minus e 1 upon k t And you have n 2 is equal to a e to the power minus e 2 upon k t So now I can find out what is n 1 upon n 2 So what is n 1 upon n 2 n 1 upon n 2 is a a gets cancelled and then you are left with e to the power minus e 1 minus e 2 Upon k t now what is e 1 minus e 2 e 1 minus e 2 is essentially equal to the frequency nu By the expression e 1 minus e 2 is equal to h nu all right So e 1 minus e 2 essentially is equal to h nu So I can substitute h nu here to get this particular expression e to the power minus h nu upon k t So if I substitute this here ok if I substitute this here Then I end up getting finally this particular expression that U which is the energy density is equal to a 2 1 upon b 2 1 is equal to b 1 2 upon b 2 1 times n 1 upon n 2 which is equal to e to the power minus h nu upon k t minus 1 so this is the energy density As predicted by Einstein Wait a minute. I just made a small mistake. So here e 2 Minus e 1 is equal to h nu So therefore this should be e 1 minus e 2 should be minus h nu so minus minus becomes plus So this should be a plus here okay, so e to the power h nu upon k t So this is what we have obtained to be The energy density of a material or a black body or a cavity as Predicted by Einstein now we can make a comparison of this with the theoretically successful expression of the Planck's Energy density distribution. So if you make a comparison with the Planck's energy density distribution, then what do we find? If we make a comparison with the Planck distribution Then the Planck's distribution is simply u nu is equal to 8 pi h nu cube upon c cube times 1 upon e to the power h nu upon k t Minus 1 so this is the distribution Given by Planck so if we make a comparison of what we obtained using the Einstein's approach with that of the Theoretically successful Planck distribution, we can immediately say from here that let me rub the board first So basically this term b 1 2 upon b 2 1 if we make a comparison is equal to 1 So that's why we have b 1 2 is equal to b 2 1 that means b 1 2 is a probability of Stimulated absorption and b 2 1 is a probability of stimulated emission and their probabilities are exactly equal number one and number two we have a 2 1 upon b 2 1 is equal to this particular expression 8 pi h which are constants upon c cube which is a constant and then you have nu Q bit h pi 8 pi h upon c cube times nu cube so if we know b 2 1 which is the stimuli probability of stimulated Emission we can automatically find a 2 1 which is a probability of spontaneous Emission so there are three distinct conclusions we can make from this Einstein's approach number one the probability of stimulated emission b 2 1 is Non-zero, okay, it is non-zero that means this proves number one that stimulated emission occurs from a theoretical perspective and also the probability of stimulated emission is exactly equal to the probability of stimulated absorption next the probability of spontaneous emission is Directly proportional to nu cube or you can say the ratio of the probability of spontaneous emission upon Stimulated emission is directly proportional to nu cube so as the energy level difference increases spontaneous emission is going to dominate over the stimulated emission all right because it is proportional to nu cube as The energy level difference increases spontaneous emission will dominate over the stimulated emission and also if we figure out any one of these probabilities We can figure out the other probabilities based on these particular expressions So this is the Einstein's approach using the Einstein coefficients to explain of course come up with an Explanation for the Planck distribution for the energy density of a material and also give us an idea about The various kinds of probabilities associated with stimulated absorption spontaneous emission and stimulated emission So that is all for today. I hope you have understood the basics of the physics behind lasers and the concept of stimulated emission and How we can correlate that with Blackbody radiation spectrum. So that is all for today. I'm Divya Jauridas. Thank you so much This is for the love of physics. I'll see you next time. Take care. Bye. Bye